The Outer Product: All of Space onto One Line

Two arrows: the blue u and the green v. Their outer product, u times v transpose, is a three by three matrix, and every column of it is a multiple of u.

Let it act on the black arrow x. First, x casts a shadow on the line of v, and that shadow gives the number v transpose x. Then this number stretches u, and out comes the orange arrow.

The rest of the narration is in the book's Companion.